Finite Group Databasebeta

← #28346991

Subgroups — #28346991

Order 13696.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293258C211yesmaximal, characteristic
2#20293255C211yesmaximal, characteristic
3#20293035C212yesmaximal, complemented
4#20293034C212yesmaximal, complemented
5#20293028C212yesmaximal, complemented
6#20293024C212yesmaximal, complemented
7#20292669C211yesmaximal, characteristic
8#20292663C211yesmaximal, characteristic
9C2×#16812040C211yesmaximal, characteristic
10#20292585C211yesmaximal, characteristic
11#428×C107C211yesmaximal, characteristic
12C22×C107⋊C8C2211yescharacteristic
13#16812109—21no—
14#16812108—21no—
15#16812108C2212yes—
16#16812108C2212yes—
17#16812101—21no—
18#16812100—21no—
19#16812100C2212yes—
20#16812100C2212yes—
21#16812088—22no—
22#16812086—22no—
23#16812068C2211yescharacteristic
24#16812068C2211yescharacteristic
25C2×C4⋊Dic107C2211yescharacteristic
26#16812043C2212yes—
27#16812042C2211yescharacteristic
28#16812042C2211yescharacteristic
29#16812041C2212yes—
30#16812040C2212yes—
31#16812040C2212yes—
32#16812040C2211yescharacteristic
33#16812040C2211yescharacteristic
34#16812019C2212yes—
35C8×Dic107C2212yes—
36C2×M16×C107C2211yescharacteristic
37#117×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#107×C107C2212yes—
40#106×C107C2212yes—
41C2×Dic214—22no—
42C2×Dic214—22no—
43C22×Dic107—21no—
44C2×C107⋊C8Q814yescomplemented
45C2×C107⋊C8C2311yescharacteristic
46C2×C107⋊C8C2311yescharacteristic
47C4⋊Dic107—21no—
48C4⋊Dic107—22no—
49C4⋊Dic107—22no—
50C4⋊Dic107D411yescharacteristic
51C4⋊Dic107D411yescharacteristic
52C4⋊Dic107C2312yes—
53C4⋊Dic107C2311yescharacteristic
54C4⋊Dic107C2311yescharacteristic
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59C22⋊Dic107—21no—
60C22⋊Dic107D412yes—
61C4×Dic107—21no—
62C4×Dic107C2312yes—
63C22×C4×C107—21no—
64C22×C4×C107C2311yescharacteristic
65M16×C107Q814yescomplemented
66C2×C8×C107C2312yes—
67C4⋊C4×C107—21no—
68C4⋊C4×C107C2312yes—
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2311yescharacteristic
71C22⋊C4×C107—21no—
72C42×C107—21no—
73C23×C107C2×D411yescharacteristic
74C2×C4×C107—21no—
75C2×C4×C107—21no—
76C2×C4×C107—22no—
77C2×C4×C107—81no—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×Q812yes—
80C2×C4×C107C2×Q812yes—
81C2×C4×C107C2411yescharacteristic
82C2×Dic107—21no—
83C2×Dic107—21no—
84C2×Dic107—22no—
85C2×Dic107—81no—
86C2×Dic107C2×D412yes—
87C2×Dic107C2×D412yes—
88Dic214—82no—
89Dic214—82no—
90C8×C107C2×Q814yes—
91C107⋊C8C2×Q812yes—
92C107⋊C8C2×Q812yes—
93C22×C107—21no—
94C22×C107—21no—
95C22×C107C4⋊Q812yes—
96C22×C107C22×D411yescharacteristic
97C4×C107—41no—
98C4×C107—42no—
99C4×C107—41no—
100C4×C107C4⋊Q812yes—
101C4×C107C22×Q811yescharacteristic
102C4×C107C22×Q811yescharacteristic
103Dic107—22no—
104Dic107—22no—
105Dic107—41no—
106Dic107—42no—
107Dic107—41no—
108C2×C107—22no—
109C2×C107C2×C4⋊Q811yescharacteristic
110C2×C107#57711yescharacteristic
111C2×C107#57711yescharacteristic
112#3163—1071nomaximal, Sylow
113C107#316311yescharacteristic, complemented, Sylow
114#544—1071no—
115#544—1071no—
116#501—1072no—
117#501—1072no—
118#499—1072no—
119#498—1072no—
120#428—1071no—
121#428D10711yescharacteristic
122#427—1071no—
123C2×#106—1071no—
124#405—1071no—
125C2×M16C2×D10711yescharacteristic
126C22×C8—1071no—
127C4⋊Q8—1072no—
128C4⋊Q8—1072no—
129C4⋊Q8—2141no—
130C4⋊Q8—2141no—
131#125—1072no—
132#125—1072no—
133#125—2141no—
134#125—2141no—
135C22⋊Q8—2142no—
136C22⋊Q8—2142no—
137#117—1071no—
138#117—1071no—
139#117C2×D10711yescharacteristic
140C2×C4⋊C4—1071no—
141C2×C4⋊C4C2×D10711yescharacteristic
142#107—1072no—
143#107—1071no—
144#107—1071no—
145#107C2×D10712yes—
146#106—1072no—
147#106—1072no—
148#106—1072no—
149#106—1071no—
150#106—1071no—
151#106C2×D10712yes—
152#97—1072no—
153C4×C8—1072no—
154C2×Q8—2142no—
155C2×Q8—2142no—
156C22×C4—21no—
157C22×C4—2141no—
158C22×C4C22×D10711yescharacteristic
159M16Dic21414yescomplemented
160C2×C8—1074no—
161C2×C8—1071no—
162C2×C8—1071no—
163C2×C8C22×D10712yes—
164C4⋊C4—21no—
165C4⋊C4—1071no—
166C4⋊C4—1071no—
167C4⋊C4—1072no—
168C4⋊C4—1071no—
169C4⋊C4—1071no—
170C4⋊C4—2142no—
171C4⋊C4—2142no—
172C4⋊C4—2142no—
173C4⋊C4—2142no—
174C4⋊C4—2142no—
175C4⋊C4—2142no—
176C4⋊C4—2141no—
177C4⋊C4C22×D10712yes—
178C4⋊C4C22×D10711yescharacteristic
179C4⋊C4C22×D10711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—1072no—
182C22⋊C4—2141no—
183C42—21no—
184C42—1072no—
185C42—2141no—
186C23D4×D10711yescharacteristic
187Q8—8562no—
188Q8—8562no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—8561no—
194C2×C4—1072no—
195C2×C4—1072no—
196C2×C4—2141no—
197C2×C4—2141no—
198C2×C4—2142no—
199C2×C4D4×D10711yescharacteristic
200C2×C4Q8×D10712yes—
201C2×C4C2×Dic21412yes—
202C2×C4C23×D10711yescharacteristic
203C8—1072no—
204C8—1072no—
205C8C2×Dic21414yes—
206C22—21no—
207C22—21no—
208C22#1681210812yes—
209C22C2×D4×D10711yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—4282no—
214C4—4281no—
215C4—4281no—
216C4—2142no—
217C4—2142no—
218C4#1681210812yes—
219C4C2×Q8×D10711yescharacteristic
220C4C22×Dic21411yescharacteristic
221C2—22no—
222C2C2×#1681210811yescharacteristic
223C2#2029343611yescharacteristic
224C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#428D107
hypocenterC107#3163
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20293035C2semidirect
4. #20293034C2semidirect
5. #20293028C2semidirect
6. #20293024C2semidirect
44. C2×C107⋊C8Q8semidirect
65. M16×C107Q8semidirect
113. C107#3163semidirect
159. M16Dic214semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202932582
#202932552
#202930352
#202930342
#202930282
#202930242
#202926692
#202926632
C2×#168120402
#202925852
#428×C1072
#3163107